SME
Grade 9 Math · Problem type
Recursive rule
Use this for “a₁ = 2 and each term is 3 times the one before, minus 1. Find a₄.”
Chart
Terms: 2, 5, 14, 41 (each term 3 × the one before − 1).
a₄ = 3 × 14 − 1 = 41
This is an example. Type your own values to start a new problem.
a₁
=
a₁
aₙ
=
k
aₙ₋₁
+
(
c
)
a
=
aₙ
Formulas
aₙ
=
term
n
of
a₁
,
each
term
k
×
the
one
before
+
c
41 = term 4 of 2, each term 3 × the one before + (−1)
Assumptions
•
A recursive rule gives the first term and how each term comes from the one before.
•
Here aₙ = k × aₙ₋₁ + c: multiply the term before by k, then add c.
•
Work out the terms in order: the chart draws each term from the one before.
Step-by-step
WE KNOW
a₁ = 2, k = 3, c = −1, n = 4
FIND
nth term (aₙ)
Step 1 · Find nth term (aₙ)
Use aₙ = term n of a₁, each term k × the one before + c
Start
at
a₁
and
use
the
rule
again
and
again:
each
term
is
k
times
the
one
before,
plus
c
.
aₙ
=
k
×
aₙ₋₁
+
c
a₂ = 3(2) − 1 = 5
a₃ = 3(5) − 1 = 14
a₄ = 3(14) − 1 = 41
aₙ = 41 → a₄ = 41
Check
Put the numbers back in. Both sides should match.
41 = 3 × 14 + (−1) ✓
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Geometric sequence: the nth term
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Adding, subtracting and multiplying polynomials
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